21 if 7 4 5 g t dt then 7 4 2 6 g t dt a 15 b 16 c 22

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21. If 4 5 g t dt , then 4 2 6 g t dt 7 7 (A) 15 (B) 16 (C) 22 (D) 23 (E) 28
AP Calculus Review 6: Area and Volume 1. The function f is continuous on the interval [4, 16]. The values of f at selected values of x are shown in the table below. Use a trapezoid rule to estimate the area under the graph of f on the interval [4, 16]. x 4 6 9 13 16 f x 8 11 15 12 10 2. Find the area bounded by the graph of 2 20 6 10 f x x x and the line y = 4. 3. Let the region R in Quadrant I be bounded by the graphs of 3 1/3 y k x and 2 2 x y k where k is a positive constant. a. Find the area of R b. Find the rate of change of the area of R with respect to c. Suppose k is decreasing at a rate of 0.2. Find the rate of change of the area of R when k d. Let k = 2. Find the value of b such that the line y = b divides the region R into 2 equal pieces. 4. Let R be the region in the first quadrant bounded by the graph of y = 8 – x 3 and the coordinate axes. a. Find the volume that results when R is rotated about the x -axis. b. Find the volume that results when R is rotated about the y -axis. c. Find the volume that results when R is rotated about the line y = 8. d. Let R be the base of a solid. Each cross section perpendicular to the x -axis is a rectangle with its base in the x - y plane and its height given by h ( t ) = x 2 . Find the volume of this solid. 5. Let the first quadrant region enclosed by the graph of 1 y x and the lines x = 1 and x = 4 be the base of a solid. If cross sections perpendicular to the x -axis are semicircles, the volume of the solid is . k . = 2. 3 4
6. The volume obtained when the region in the first quadrant bounded by the x -axis and the curve y = 2 x is revolved about the x -axis is x 2 8
7. The region in the first quadrant bounded by the graph of y = arcsin x , y = /2 and the y -axis is rotated around the y-axis. The volume of the resulting solid is given by 2 2 2 2 2
8. Let the base of a solid be the first quadrant region enclosed by the x -axis, the y -axis and the graph of 2 1 4 x y   . If all cross sections perpendicular to the y -axis are squares, then the volume of the solid is (A) 3 (B) 2 (C) 1 (D) 1/2 (E) 1/3
AP Calculus Review 7: Calculus Applications 1. Find the average value of 3 2 8 12 f x x x x  on the interval in the first quadrant where 0 f x 2. Find the value of k such that the average value of f x x on the interval [0, k ] is 2. 3. A particle moves along the x -axis with a velocity given by 2 sin 12 t v t t for 0 ≤ t ≤ 6. At time t = 0, the particle was at x = a. What was the particle’s distance traveled from t = 0 to t = 6 ? b. What was the particle’s position at t = 3 ? . 2.

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